Introduction to Algebraic Topology · Simplicial homology is easy to compute (for easy examples). ’Abelianising’ the fundamental group of a space π1(X) gives the ﬁrst homology

∆-complexesSimplicial homology and cohomology

Simplicial homology with closed supportPoincare Duality

Long exact sequences

Introduction to Algebraic Topology

Nir Avni and Jesus Martınez Garcıa

Sheaves in Representation Theory, Skye

24 May, 2010

Nir Avni and Jesus Martınez Garcıa Introduction to Algebraic Topology




Motivation (I)

Simplicial homology groups “measure” n-dimensional holes intopological spaces.

It provides “global” information.

It is a topological tool, mainly used in topology, but also ingeometry and algebra.

There are other definitions for (co)homology (Morse, singular,deRham, quantum, sheaf...) and (often) isomorphisms amongthem.





Motivation (II)

Simplicial homology is easy to compute (for easy examples).

’Abelianising’ the fundamental group of a space π1(X ) givesthe first homology group of X .

An orientation-free approach requires more computations(triangulations [Mun84]).





∆-complexes

Definition

Let {a0, . . . , an} be a geometrically independent set in Rn. The

n-simplex ∆n spanned by a0, . . . , an is:

∆n = {x =n

∑

i=0

tiai |n

∑

i=0

ti = 1}.





The subsimplices obtained by removing one single vertex are calledfaces (there are n + 1 of them). The union of all faces is ∂∆n.∆n has a topology induced by the usual topology of Rn. The open

simplex of ∆n is◦

∆n:= ∆n \ ∂∆.

Definition

A ∆-complex structure on a topological space X is a collection ofmaps ∆X = {σα : ∆n −→ X, n depending on α}, such that:

(i) σα| ◦

∆nis injective ∀α, and all x ∈ X is in the image of one and

only one such restriction.

(ii) Each restriction of σα to a face of ∆n is a mapσβ : ∆n−1 −→ X , σβ ∈ ∆X .

(iii) A set A ⊆ X is open iff σ−1α (A) is open ∀σα.





Examples of ∆-complexes: RP2 and S1





DefinitionsComputations

Simplicial chain complex

Definition

Let Cn(X ) be the free abelian group with basis the openn-simplices enα of X . Its elements σα =

∑

α nαenα, are called

n-chains.The boundary homomorphism ∂n : Cn(X ) −→ Cn−1(X ) is definedon basis elements as:

∂n(σα) =∑

i

(−1)iσα|[v0,...,vi ,...,vn].

Lemma ([Hat02]: lemma 2.1)

The composition Cn+1(X )∂n+1−→ Cn(X )

∂n−→ Cn−1(X ) is zero.






Figure: Boundaries of the standard simplices. [Hat02]

Definition

If for σ ∈ Cn(X ), ∂n(σ) = 0, then σ is a cycle. If σ = ∂n+1(σ),then σ is a boundary.






Definition

The n-th simplicial homology group of X is:

Hn(X ) =Ker ∂n

Im ∂n+1.

The n-th betti number of X is the rank of Hn as a Z-module:

bn(X ) = rkZ(Hn(X )).

Lemma

The definition does not depend on the ∆-complex structure chosenfor X .






Cohomology

Definition

Consider the dual, cochain group Cn(X ) := Hom(Cn(X ),Z). Wealso have a coboundary homomorphism δn : Cn−1 −→ Cn defined,for γ : Cn(X ) −→ Z, by its action:

δ(γ)(σ) := γ(∂(σ)).

Now, we have cocycles (δ(σ) = 0) and coboundaries (σ = δσ) and,since δ2 = 0, the cohomology groups:

Hn(X ) =Ker δn

Im δn+1.






Affine space

If f : X −→ Y is continous, we get a f♯ : Cn(X ) −→ Cn(Y ) and∂ ◦ f = f ◦ ∂. Therefore we have f∗ : Hn(X ) −→ Hn(Y ).

Theorem

If f , g : X −→ Y are homotopic then f∗ = g∗ and therefore if X ,Yare homotopic, H∗(X ) = H∗(Y ).

Hence, since ∆0 ∼= Rk , H∗(R

k) = H∗(∆0). Its chain complex is:

0 → 0 → · · · → C0 = Z → 0.

And:

Hn(Rk) =

{

Z n = 0;0 n 6= 0.





Borel–Moore Homology

Simplicial Homology With Closed Support is defined similarlyto homology with compact support, but allowing the chains to beinfinite combinations of simplices.

There are both more cycles and more boundaries.

In the case where X has a finite triangulation, HBM

k(X ) = Hk(X ).





Affine space

Suppose that σ =∑

an[n] is a cycle in R.

Define τ =∑

bn[n, n+1] by setting b0 = 0, and setting bn so thatthe coefficient of [n − 1] in ∂τ is 0. We get that σ is a boundary.

⇒ HBM

0 (R) = 0





Affine space

The chain σ =∑

[n, n + 1] is a cycle. Conversely, any 1-cycle is amultiple of σ.

HBM

1 (R) = K

Similarly, one shows

HBM

k(Rn) =

{

0 k 6= nK k = n





Poincare Duality

If X is a smooth, n-dimensional, orientable manifold, then anyfine-enough triangulation has a dual





Poincare Duality

The vertices of the dual correspond to the top simplices of theoriginal triangulation.

The edges of the dual correspond to the facets of the originaltriangulation....

The top cells of the dual correspond to the vertices of theoriginal triangulation.

⇒ CBM

i (X ) = Cn−i (X )∨

In fact, HBM

i(X ) = Hn−i (X )∨.





LES for a closed set and its complementComputation: Complex Projective Space

LES for a closed set and its complement

Suppose that X is a triangulated space, and let F ⊂ X be asub-complex. Let U = X \ F , and triangulate it by dividing eachopen simplex into (infinitly many) sub-simplices.







If σ is a k-cycle in F , then it is a k-cycle in X . We get a mapBMk(F ) → BMk(X ).

If τ is a k-cycle in X , then its restriction to U can be presented asa cycle. We get a map BMk(X ) → BMk(U).

If ξ is a k-cycle in U, then its boundary in X is a cycle in F . Weget a map BMk(U) → BMk−1(F )







We get a sequence

· · · → HBM

k(F ) → HBM

k(X ) → HBM

k(U) → HBM

k−1(F ) → HBM

k−1(X ) → · · ·

which is exact.






Computation: Complex Projective Space

Pn = C

n ⊔ Pn−1. Looking at the LES of open/closed subsets, we

get

→ HBM

k+1(Cn) → HBM

k(Pn−1) → HBM

k(Pn) → HBM

k(Cn) → HBM

k−1(Pn−1) →

If k 6= 2n, 2n − 1 then BMk(Pn) = BMk(P

n−1). By induction,

BM2n(Pn−1) = BM2n−1(P

n−1) = 0. So if k = 2n, thenBM2n(P

n) = BM2n(Cn) = K , Whereas if k = 2n − 1, then

BM2n−1(Pn) = 0.






References I

Allen Hatcher.Algebraic topology.Cambridge University Press, Cambridge, 2002.

James R. Munkres.Elements of algebraic topology.Addison-Wesley Publishing Company, Menlo Park, CA, 1984.


Documents

Introduction to Algebraic Topology · Simplicial homology is easy to compute (for easy examples). ’Abelianising’ the fundamental group of a space π1(X) gives the ﬁrst homology